A Kinetic Monte Carlo StudyOf Ordering in a Binary Alloy

Group 3:

Tim Drews (ChE)

Dan Finkenstadt (Physics)

Xuemin Gu (MSE)

CSE 373/MatSE 385/Physics 363 Final Project

University of Illinois at Urbana-Champaign

December 14, 2000

Code Introduction

• Main part of project: development of Kinetic Monte Carlo (KMC) code to simulate ordering in a binary superalloy

• Equilibrium simulations that compute a vacancy pathway through the bulk alloy• Simulated alloy: similar to Fe-Al

1. Exhibits three phase behavior: 2. B2 ordered phase3. A2 disordered phase4. A2+B2 mixed phase

• Compared the results to phase diagrams from the literature

• Developed data analysis code to compute various order parameters

• Developed many visualization techniques to determine the phase of the alloy

Kinetic Monte Carlo Method

Move

Nearest Neighbor Sites0 1

1 2 3 4 5 6 7 8

Jump Frequency and Potential Function

kT

Eexp

actXV

XV

2,1i

ixxi

iyi

actXV )nun(

2

1E

1j

1i

j

1iii )c(R)c(

)c(

texp)t,c(P

18

1ii )c()c(

where

Phase Diagrams

Theoretical Simulated with GrandCanonical MC Simulations

Reproduced from A Monte-Carlo Study of B2 Ordering and Precipitation Via Vacancy Mechanism in B.C.C. Lattices. Athenes, M., Bellon, P., Martin, G., and Haider, F.

Acta Metallurgica. Vol. 44, No. 12, pp. 4739-4748, 1996.

BCC Lattice

Bulk BCC Lattice Two Simple Cubic Lattices

sublattice(arbitrary)

sublattice(arbitrary)

64 Cell Cubic Lattice - Disordered A2 Phase

T = 1000 K u1 = -0.041*107 MC Steps cB = 0.25

sublattice sublattice 2 plot

32 Cell Cubic Lattice - High and Low Order B2 Phase

sublattice sublattice 2 plot

T = 700 KMC steps = 1*107

u1 = -0.04cB = 0.25

T = 700 KMC steps = 1*107

u1 = -0.04cB = 0.50

64 Cell Cubic Lattice - High Order B2 Phase

sublattice sublattice 2 plot

T = 700 K u1 = -0.041*107 MC Steps cB = 0.45

16, 32, and 64 Cell Cubic Lattice - High Order B2 Phase

sublattice sublattice sublattice sublattice

sublattice sublattice

1*107 MC steps

1*106 MC stepsT = 700 Ku1 = -0.04cB = 0.45

32 Cell Cubic Lattice - Mixed A2+B2 Phase

sublattice sublattice 2 plot

T = 200 K u1 = -0.041*107 MC Steps cB = 0.25

Conclusions

• KMC code can generate expected equilibrium phase behavior, for a given temperature and concentration, provided enough MC steps are taken for the given simulation cell size

• Observed dynamic growth and ageing

• Did not distinguish phase transitions• Could do this by computing total free energy and looking for kinks in the free energy diagram• Could also look for critical slowing down or finite-size scaling phenomena

• Did not vary the activation energy• Done by Athenes, et al., and this energy was found to have a significant effect• Can be readily done with this code